Once galaxies settle their discs and become self-gravitating, stellar bars can form, driving the subsequent evolution of their host galaxy. Determining the ages of bars can therefore shed light on the epoch of the onset of secular evolution. For this work we applied the first broadly applicable methodology to derive bar ages to a sample of 20 nearby galaxies. The method is based on the co-eval build-up of nuclear structures and bars, and involves using integral field spectroscopic (IFS) data from the Multi Unit Spectroscopic Explorer (MUSE) instrument on the Very Large Telescope to disentangle the star formation history of the nuclear disc from the background population. This allows us to derive the formation epoch of the nuclear disc, and thus of the bar. We estimated the bar formation epoch of nearby galaxies, mostly from the TIMER survey, and have created the largest sample of galaxies with known bar ages to date. We find bar formation epochs varying between 1 and 13 Gyr ago, illustrating how disc-settling and bar formation are processes that first took place in the early Universe and are still taking place in some galaxies. We infer the bar fraction over cosmological time with our sample, finding remarkable agreement with that obtained from direct studies of galaxies at high redshifts. Additionally, for the first time, we were able to investigate secular evolution processes taking into account the ages of bars. Our results agree with the scenario in which bars aid the quenching of the host galaxy; galaxies that host older bars tend to be more quenched. We also find that older bars tend to be longer and stronger, and to host larger nuclear discs. Furthermore, we find evidence that the nuclear disc stellar mass builds up over time. On the other hand, we find no evidence that downsizing plays a role in bar formation since we find that bar age is independent of galaxy stellar mass. With the means to estimate bar ages, we can begin to understand better when and how bars shape the observed properties of disc galaxies.